Dynamic Phasors for Small Signal Stability Analysis

Transcription

1 for Small Signal Stability Analysis Chandana Karawita (Transgrid Solutions) for Small Signal Stability Analysis

2 Outline Introduction 1 Introduction Simulation and Analysis Techniques Typical Outputs Modelling of Components 2 3 Interactions Between Nearby HVDC Converters Torsional Interactions 4 for Small Signal Stability Analysis

3 Simulation and Analysis Techniques Typical Outputs Modelling of Components Power System Simulation and Analysis Electromagnetic Transient Simulation - Time Domain Technique Transient Stability Simulation - Time Domain Technique Small Signal Stability Analysis - Frequency Domain Technique for Small Signal Stability Analysis

4 Simulation and Analysis Techniques Typical Outputs Modelling of Components Typical Output of an EMT Simulation Sample responses of a four-generator power system after a three phase fault The amplitude of the 60 Hz voltage waveform is modulated by the low frequency of oscillations of the rotor. for Small Signal Stability Analysis

5 Simulation and Analysis Techniques Typical Outputs Modelling of Components Typical Output of a TS Simulation Sample responses of a four-generator power system after a three phase fault Rotor angle of generator 2 and the rms voltage of Bus 2 show low frequency oscillations around 1 Hz. Transient Simulation Generator Speed Change (Rad/s) W Min Max Diff Bus Voltage (kv) V Min Max Diff Time(s) f 1.2 for Small Signal Stability Analysis

6 Simulation and Analysis Techniques Typical Outputs Modelling of Components Typical Output of a Small Signal Analysis Structural Information Oscillation modes (frequencies and corresponding damping). Mode Shapes of oscillation frequencies. Participation of state variables in oscillation modes. Observability of oscillation modes. Controllability of oscillation modes. Residues for input-output pairs. for Small Signal Stability Analysis

7 Simulation and Analysis Techniques Typical Outputs Modelling of Components Typical Output of a Small Signal Analysis: Participation Factors Participation Factors show the relative participation of state variables when a mode is excited Generator HVDC 1 HVDC Z2 F1 and F2 Tie 2 Percentage Participation State Variables for Small Signal Stability Analysis

8 Simulation and Analysis Techniques Typical Outputs Modelling of Components Typical Output of a Small Signal Analysis: Mode Shape Mode Shape shows whether the state variables are oscillating together or not X r2 X r1 60 I dci1 V cap I dcr2 V cap V cap2 I dcr1 V cap1 210 I 330 dcr I dcr I I 270 dci2 dci1 I dci2 Mode 6 Mode for Small Signal Stability Analysis 90 90

9 Machine Models Introduction Simulation and Analysis Techniques Typical Outputs Modelling of Components Transient Stability and Small Signal Stability Rotor fluxes are modelled as state variables. Stator fluxes are NOT modelled as state variables. Electromagnetic Transient Simulation Rotor fluxes and stator fluxes are modelled as state variables. Common to both Dynamics of the rotor and that of auxiliary controllers are modelled using differential equations. for Small Signal Stability Analysis

10 Transmission Line Models Simulation and Analysis Techniques Typical Outputs Modelling of Components Transient Stability and Small Signal Stability Series inductance and shunt capacitance are modelled as constant impedances (admittances) calculated at the nominal frequency ω 0. Electromagnetic Transient Simulation Transmission line is modelled using differential equations (telegraphic equations). for Small Signal Stability Analysis

11 Simulation and Analysis Techniques Typical Outputs Modelling of Components Small Signal Stability: Frequency domain technique Only accurate in the vicinity of nominal frequency. Structural Information relevant to the system is available. Transient Stability: Time domain technique Only accurate in the vicinity of nominal frequency. Large integration time step is used simulation is fast. Electromagnetic Transient Simulation: Time domain technique Accurate over a wide frequency range. Integration time step is small simulation is slow. for Small Signal Stability Analysis

12 Introduction Instantaneous Current Waveform i ac = A m e jφ e jω 0t = [A m cos(φ) + ja m sin(φ)]e jω 0t A m is the magnitude of the current, φ is the phase of the current, and ω 0 is the nominal system frequency. In Rectangular Coordinates i ac = (I R + ji I )e jω 0t for Small Signal Stability Analysis

13 Modelling a Transmission Line using Dynamic Phasors Series Branch Series R-L circuit connected between nodes 1 and 2. v 12 = L di 12 dt + Ri 12 Using the Complex rotating phasor relationships (V R + jv I )e jω 0t = L d(i R + ji I )e jω 0t dt + R(I R + ji I )e jω 0t for Small Signal Stability Analysis

14 Assuming that the nominal system frequency (ω 0 ) is constant V R + jv I = L d(i R + ji I ) dt + (R + jω 0 L)(I R + ji I ) Since L is in pu, (ω 0 /L) terms appear instead of (1/L) [ ] İ R İI = [ Rω0 ] [ ] ω 0 ω L 0 IR L 0 ω 0 0 L Rω ω I L I 0 ω 0 L 0 ω 0 L V 1 R V 1 I V 2 R V 2 I for Small Signal Stability Analysis

15 Modelling a Transmission Line using Dynamic Phasors Parallel Branch [ ] V1R V 1I = [ ω0 ] [ ] ω 0 ω 0 [ ] RC 0 V1R C ω ω 0 + IR 0 V RC 1I 0 ω I 0 I C for Small Signal Stability Analysis

16 Other Interpretations of d-q Components of Network Voltages and Currents Network voltages and currents are represented by their d-q components which are modelled as state variables. Fourier Components of Network Voltages and Currents Network voltages and currents are represented by their Fourier components which are modelled as state variables. for Small Signal Stability Analysis

17 Power System Signals as Amplitude Modulated Signals If R and I components are constants The instantaneous waveforms are sinusoidal. If R and I components are oscillating at frequency ω The instantaneous waveforms are amplitude modulated waveforms with carrier frequency ω 0. This results in two sidebands of ω 0 ω and ω o + ω for Small Signal Stability Analysis

18 Power System Signals as Amplitude Modulated Signals Example If f 0 = 60 Hz and f = 5 Hz, the two sideband frequencies are f 1 = 55 Hz and f 2 = 65 Hz. Both are close to 60 Hz and the constant admittance representation of transmission network is acceptable. Example If f 0 = 60 Hz and f = 25 Hz, the two sideband frequencies are f 1 = 35 Hz and f 2 = 85 Hz. Both are significantly different to 60 Hz and the constant admittance representation of transmission network is NOT acceptable. for Small Signal Stability Analysis

19 Interactions Between Nearby HVDC Converters Torsional Interactions Interactions Between Nearby HVDC Converters A simple Network for model Validation Two HVDC lines, ac filters, ac transmission line, and a generator. A pulse of magnitude of 5 % and duration 0.3s was applied to the rectifier current controller input. for Small Signal Stability Analysis

20 Interactions Between Nearby HVDC Converters Torsional Interactions Comparison of EMT, SS-traditional, and SS-Dynamic Phasor Approach (a) Change in Idcr of HVDC1 current (ka) PSCAD/EMTDC Model 1 Model 2 current (ka) time(s) (b) Change in Idcr of HVDC PSCAD/EMTDC Model Model time(s) for Small Signal Stability Analysis

21 Interactions Between Nearby HVDC Converters Torsional Interactions Rotor Oscillations: SS-traditional and SS-Dynamic Phasors give same results 1 x Change in Generator Speed PSCAD/EMTDC Model 1 Model 2 speed (pu) time(s) for Small Signal Stability Analysis

22 Interactions Between Nearby HVDC Converters Torsional Interactions Frequency Response of the Model EMT Vs SS-Dynamic Phasor % change phase angle (Deg.) Magnitude (input:, output: V cap ) PSCAD/EMTDC SSS model frequency (Hz) Phase (input:, output: V ) cap PSCAD/EMTDC SSS model frequency (Hz) for Small Signal Stability Analysis

23 Response to a 200 Hz signal Interactions Between Nearby HVDC Converters Torsional Interactions Changes in Rectifier side DC currents for a 5 %, 200Hz sinusoidal change of the HVDC1 rectifier side AC source voltage (VS1). (a) Change in Idcr of HVDC PSCAD/EMTDC Model 2 Model 1 current (ka) time(s) 0.01 (b) Change in Idcr of HVDC2 PSCAD/EMTDC Model 2 Model current (ka) time(s) for Small Signal Stability Analysis

24 Interactions Between Nearby HVDC Converters Torsional Interactions Participation Factors presence of an interaction between the two HVDC converters Generator HVDC 1 Z2 Z1 F1 and F2 HVDC 2 Z3 Z4 F3 and F4 Tie 90 Percentage Participation State Variables for Small Signal Stability Analysis

25 Interactions Between Nearby HVDC Converters Torsional Interactions Mode Shape state variables of the two converters oscillate against each other X r2 90 X r1 60 I dci1 150 V cap I dcr2 V cap I dcr1 I dcr1 210 I dcr I dci2 I 270 dci1 I dci2 Mode 6 Mode for Small Signal Stability Analysis 90

26 Interactions Between Nearby HVDC Converters Torsional Interactions Torsional Interactions The CIGRE benchmark HVDC test system with some modifications. A synchronous generator is connected at rectifier side AC bus to supply half of the P-Q requirement of rectifier. S1 Z1 Rectifier DC Line Inverter Z2 S2 Idcr Vcap Idci G Generator with a multi-mass turbine F1 F2 for Small Signal Stability Analysis

27 Interactions Between Nearby HVDC Converters Torsional Interactions Comparison of EMT and SS-Dynamic Phasor SS-Dynamic-Phasor provides accurate results in the frequency range of interest 10 % change in rectifier current reference for 10 ms Change in Rectifier side DC current PSCAD/EMTDC SSS model 0.08 current (ka) time(s) for Small Signal Stability Analysis

28 Torsional Interaction Modes Interactions Between Nearby HVDC Converters Torsional Interactions Mode Freq. D Major Participants (Hz) (%) A HVDC-Generator-Turbine B HVDC-Generator-Turbine for Small Signal Stability Analysis

29 Interactions Between Nearby HVDC Converters Torsional Interactions Participating states are identified using Participation Factors 100 Mode A Gen Turbine HVDC Mode B !!! LPB LPA! IP! gen HP X CCC X I CEA dcr I dci V cap for Small Signal Stability Analysis

30 Publications Introduction Interactions Between Nearby HVDC Converters Torsional Interactions 1 C. Karawita, U.D. Annakkage, A Hybrid Network Model for Small Signal Stability Analysis of Power Systems, IEEE Transactions on Power Systems, Vol 25, 2010, Page(s): C. Karawita, U.D. Annakkage, Multi-Infeed HVDC Interaction Studies Using Small-Signal Stability Assessment, IEEE Transactions on Power Delivery, Vol 24, 2009, Page(s): C. Karawita, U.D. Annakkage, HVDC-Generator-Turbine Torsional Interaction Studies Using A Linearized Model With Dynamic Network Representation, International Conference on Power System Transients (IPST), Kyoto, Japan, June for Small Signal Stability Analysis

31 SSR between DFIG based Wind Power Plant and series compensated transmission lines (Hiranya). SSI between nearby LCC-HVDC and VSC-HVDC terminals (Kevin MH). SSR mitigation using FACTS controllers (TGS). Transient Stability Simulation using (Rae MH). Chandana has developed an SSR Small Signal Analysis Program for Small Signal Stability Analysis

32 Acknowledgements Research Funding HVDC Interactions NSERC, University of Manitoba, and Province of Manitoba. Research Funding Wind Power Plant SSR Studies NSERC and Manitoba Hydro. Valuable Feedback Bret Davis, Ioni Fernando, Ani Gole, Shaahin Filizadeh, and Garth Irwin. for Small Signal Stability Analysis